is the \(\phi \)  and \(k\)  dependent expectation value.
In the limit \(k\rightarrow 0\)  the cut-off vanishes and EAA reduces to the one-part irreducible effective action \(\Gamma [\phi ]= \Gamma _0[\phi ]\)  for the field \(\hat{\phi }_0\) .
In the opposing limit \(k \rightarrow \infty \)  the EAA reduces to the bare action written in terms of the fields \(\hat{\phi }_\infty \) .
Let us note that we could additionally make a change of integration variables in the RHS of (REF ) which keeps \(\Gamma _k[\phi ]\)  invariant provided we make this change everywhere including in the measure. Here we are keeping the integration variables \(\hat{\chi }\)  and the bare action \(S[\hat{\chi }]\)  \(k\) -independent, such that the \(k\) -dependence comes only from the regulator and the composite fields \(\hat{\phi }_k[\hat{\chi }]\) . Since we are ultimately interested in computing observables (REF ) at vanishing regulator and on the equations of motion for \(\Gamma _0[\phi ]\)  we will recover (REF ) independently of the regulator and the parameterisation \(\hat{\phi }_k[\hat{\chi }]\) . For more details we refer the reader to [1]}.
